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THE GEOMETRY OF CIRCLES ^ 



ORTHOGONAL TO A GIVEN SPHERE 



BY 

CHARLES SAVAGE FORBES 



Submitted in Partial Fulfilment op the Requirements for the Degree 

of Doctor of Philosophy in the Faculty of Pure Science 

Columbia University 



NEW YORK 
1904 



^ 



THE GEOMETRY OF CIRCLES 
ORTHOGONAL TO A GIVEN SPHERE 



CHARLES SAVAGE FORBES 



Submitted in Partial Fulfilment of the Requirements for the Degree 

of Doctor of Philosophy in the Faculty of Pure Science 

Columbia University 



NEW YORK 

1904 



ft 



£ 36f/^ 




3- DEC- 6 



TABLE OF CONTENTS 



PAGE. 

Introductory Considerations 1 

CHAPTER I 

Introductory Theorems upon Orthogonal Spheres and Circles 

Dimensionality 2 

Definitions of Orthogonality 2 

General Theorems 2 

Circle Cut out of Sphere by Plane 3 

Intersections of Orthogonal Spheres are Orthogonal Circles 4 

Dimensionality of Assemblage 6 

Circle Fixed by Two Points 6 

Corresponding Points 7 

Summary . 9 

CHAPTER II 

Coordinates of the Circle 

A. Pentaspherical Coordinates . . 9 

B. Point-pairs 11 

4 

C. Dual Interpretation of ~^a.x. = 13 

D. Coordinates of a Circle as Envelope of Spheres 14 

E. Coordinates of a Circle as Locus of Point-pairs 15 

F. Condition that Two Circles Intersect 16 

G. Conditions that a Given Circle may Lie on a Given Sphere or Pass through 

a Given Point-pair 17 

CHAPTER III 

The Linear Complex op Circles 



A. Linear Complex of Circles Regarded as Envelopes of Spheres 

B. Linear Complex of Circles Regarded as Loci of Point-pairs 

C. Polar Spheres and Pole Point-pairs 

D. Second Proof of Fundamental Correlation 

E. Comparison between Circle Geometry and Associate Theories 



. 18 
. 19 
. 20 
. 21 



IV TABLE OF CONTENTS 

CHAPTER IV 

Conjugate Circles, Pole Point-pairs and Polar Spheres 

A. General Theorems . . . . . . . . . . .24 

B. Special Complexes of Circles 25 

C. Coordinates of Conjugate Circles ......... 26 

D. Construction of the Complex , ... 28 

CHAPTER V 
Linear Congruences of Circles and Problems upon the Assemblage 

A. The Linear Congruence 30 

B. The Surface of Intersection of Two Congruences of a Complex . . .32 

C. Radius of a Circle, Circles in Involution, Degeneration of *S 5 . . . .33 

CHAPTER VI 
Families of Spheres 36 

CHAPTER VII 

Transformations of the Assemblage 

A. Generalization of Coordinates y ik ......... 40 

B. Transformation of Fundamental Spheres 41 

C. Invariance of <•> ( A ) 43 

D. Projective Transformations by Means of Complexes 44 

CHAPTER VIII 

The Complex of Second Degree' 

A. Tangent and Polar Linear Complexes 46 

B. Intersection of a Pencil of Circles with C 2 48 

C. Generalization of Pliicker 49 

BIBLIOGRAPHY . . . . . .49 

INDEX iii 



THE GEOMETRY OF CIRCLES ORTHOGONAL TO A GIVEN SPHERE 

BY 
CHARLES SAVAGE FORBES 

Introductory Considerations 

It is well known that the progress o£ geometry in the past century has been 
due in large measure to recognition of the fact that the choice of element is 
arbitrary. To one trained in the Pliicker line geometry, the conception of space 
as an assemblage of lines becomes both convenient and natural. We have 
chosen the circle as an element because in the domain to which we invite the 
reader's attention the circle is the natural element. We treat of the assemblage 
of circles orthogonal to a given sphere. At the outset we regard this system as 
a part of ordinary space, but once the foundations are laid we shall think and 
move in a space of circles. 

Koenigs has roughly outlined the six dimensional theory of circles in space. 
Our space is related to his much as the plane is related to ordinary point space. 
We have restricted our domain, but have thereby rendered possible a more 
searching inquiry. The geometry of circles orthogonal to a given sphere is 
closely related to the Pliicker line geometry. Both are four-dimensonal, and 
like the geometry of the planes of a point in space of four dimensions each 
possesses a self-reciprocal element. 

The circle is dually regarded as an envelope of spheres, or as a locus of point- 
pairs. The sphere and the point-pair are reciprocal notions, and the properties 
of configurations of circles are stated dually in terms of these fundamental con- 
cepts. Two sets of six circle coordinates arise, and a linear relation between 
either set defines a complex of circles. Point-pairs and spheres are correlated 
by each complex as pole and polar, and either contains all the circles of the 
complex belonging to the other. Hence the circles of a complex lying upon a 
sphere constitute a pencil of circles. Each circle has a conjugate circle with 
respect to every complex. Two complexes intersect in a congruence and three 
complexes in a circle surface, closely related to the hyperboloid of one sheet. 
The geometry of families of spheres is found similar to that of systems of parallel 
planes. The linear transformation of the spheres of reference, the transforma- 
tion by inversion, and the projective transformations are utilized ; and there is 
set up a geometry of polar and tangent linear complexes with respect to a com- 
plex of second degree. 

1 



Z C. S. FOBBES: THE GEOMETRY OF CIECLE8 

I am indebted to Professor C. J. Keyser of Columbia University for the 
suggestion of the topic and much kindly criticism. 



CHAPTER I 

Intboductoey Theobems upon Obthogonal Sphebes and Cibcles 

In this chapter we establish certain elementary theorems upon which the sub- 
sequent theory is based. 

Theobem I. Tliere are oo 3 planes in three dimensional space. 
The equation of a plane is 

Ax + By + Cz + 1 = , 

which involves three parameters, and gives to the plane three degrees of freedom. 

Theobem II. There are oo 6 circles in three dimensional space. 

The equation of a circle in a plane involves three parameters. No two planes 
have a circle of finite radius in common. Hence (Theorem I), there are 

oo 3 X oo 3 = oo 6 
circles in space. 

Theobem III. There are go 4 spheres in three dimensional space. 
The equation of a sphere, 

x 2 + y i + z 2 + zax + 2by + 2cz + d=0 

involves four parameters. 

Definition I. Ttoo spheres are orthogonal, when the square of the distance 
of their centers is equal to the sum of the squares of their radii. 

Theobem IV. There are oo 3 spheres orthogonal to a given sphere. 

Definition I imposes one condition upon the four parameters of the arbitrary 
sphere, leaving to it three degrees of freedom. 

Definition II. Two intersecting circles are orthogonal when their tangents 
at the point of intersection are perpendicular. 

Definition III. A circle is orthogonal to a sphere when it is orthogonal to 
every great circle of the sphere passing through one of the points of intersec- 
tion. 

Theobem V. A circle orthogonal to two great circles of a sphere at their 
point of intersection is orthogonal to the sphere. 

For the tangents to the great circles through the point of intersection all lie 
in a plane tangent to the sphere at the point. The tangent of the circle being 
perpendicular to two of these tangent lines is perpendicular to the plane and 
therefore to them all. Furthermore this tangent line being perpendicular to a 



ORTHOGONAL TO A GIVEN SPHERE 3 

tangent plane at the point of tangency goes through the center of the sphere. 
A circle and its tangent line lie in a same plane. From this follows : 

Theorem VI. Every circle orthogonal to a sphere, lies in a plane passing 
through the center of the sphere. 

Theorem VII. Every sphere passing through a circle orthogonal to a 
sphere is orthogonal to that sphere. 

Let C be a circle orthogonal to a sphere S. Pass a plane II through C and 
point O the centre of 8 (Theorem VI). II cuts out of S a circle C, orthogonal 
to circle C (Definition III). Let A be the centre of circle C. Let P be one 
of the points of intersection of circles C and C . Draw the tangents OP and 
AP to C and C. These tangents are perpendicular. Pass any sphere S' 
through circle C . The center O' of S' is in a perpendicular A 0' to the plane 
II. OP is therefore perpendicular to OP. Hence 

~OU 2 = OP 2 + OlP 2 . 

But 0' is the distance of the centres of spheres 8 and S' ; and OP and O'P 
are their respective radii. Therefore S and S' are orthogonal (Definition I). 
We next find the analytical condition (rectangular coordinates) that the spheres 

(JS) x 2 + y 2 + z 2 =R 2 , 

(£') x 2 + y 2 + z 2 + 2ax + 2by + 2cz + a 2 + b 2 + c 2 = r 2 , 

shall be orthogonal. The condition is (Definition I) 

a 2 +b 2 + c 2 =P 2 + r 2 . 
Whence 

r 2 = a 2 + b 2 + c 2 - PL 2 
and S' becomes 

x 2 + y 2 + z 2 + 2ax + 2by + 2cz = — R 2 . 
Hence the result : 

Theorem VIII. A sphere S' is orthogonal to a sphere S, whose center is 
the origin, when the absolute term, of S' is equal to the square of the radius 
ofS. 

Theorem IX. Every plane passing through the center of a sphere S, cuts 
out of a second sphere S' , orthogonal to S, a circle C, orthogonal to the circle 
C cut out of S by the plane. 
Let the equations of the spheres be 

(S) x 2 + y 2 + z 2 =P 2 , 

S') x 2 + y 2 + z 2 +2ax + 2by + 2cz=-P 2 . 



4 C. S. FORBES: THE GEOMETRY OF CIRCLES 

Since the property is geometrical, it is independent of the choice of axes. The 
cutting plane may therefore be taken as 

3 = 0. 

This gives the circles 

(C) x 2 + y 2 = P 2 , 

(C) x 2 + y 2 + 2ax + 2by=-fi 2 . 

These circles are orthogonal, for they lie in a plane, and 

a 2 +b 2 =P 2 +a 2 +b 2 -E 2 . 

That is, the square of the distance of their centers is equal to the sum of the 
squares of their radii. Hence their tangents at the points of intersection are 
perpendicular, and they are orthogonal (Definition II). 

Theorem X. Two spheres orthogonal to a same sphere intersect in a circle 
orthogonal to that sphere. 

Let the spheres 

(8') x 2 + y 2 + z 2 +2ax+2by + 2cz=-fi 2 , 

(8") x 2 + y 2 + z 2 + 2a'x + 26 'y + 2c z = - R 2 , 

be orthogonal to 

(8) x 2 + y 2 + z 2 = P 2 . 

The intersection of 8' and S" is a circle C lying in the plane 

(II) {a — a')x + (b — b')y+ (c — c')s=0. 

This plane II passes through the center, O, of sphere S. Let P be one of the 
points where circle C intersects sphere 8. Pass a plane II' through P and O, 
perpendicular to plane II . LT ' cuts out of sphere S a circle C ' orthogonal to 
C (Definition II). We have the Lemma : 

Every circle lying in a plane which passes through the center of a sphere 
is orthogonal to a great circle of that sphere at each of its points of intersec- 
tion with the sphere. 

C is also orthogonal to circle C" cut out of S by II (Theorem IX). C being 
orthogonal to two great circles C and C" of sphere S, is orthogonal to S 
(Theorem V). 

Theorem X may be analytically established as follows, whether the circle of 
intersection be real or imaginary. The equation of the fundamental sphere is 

(8) x 2 + f + z 2 = E 2 . 



ORTHOGOKAL TO A GIVEN SPHERE 5 

Any two spheres orthogonal to S are given by 

(#') x 2 + y 2 + z 2 + 2a'x + 2b'y + 2c'z + R 2 = 0, 

(S") x 2 + y 2 + z 2 + 2(a + a')x + 2(6 + h')y + 2(c + c')z + R 2 = 0. 

The axial plane of S' and S" is 

( II ) ax + by + cz = , 

and of S and /S" is 

(IT) ax + b'y + c'z + R 2 = . 

The intersections of II , II ' and S are the points of intersection of the three 
spheres, i. e., the points where the circle ( S', S") pierces S. Solving simul- 
taneously for x', y' and z the coordinates of these points we obtain, 

, _ [c(c'a - ca' ) + b{b'a — la' )~\R 2 ± R(c'b - cb' )VT?, 

,_ [a(a'b-ab') + c(c' b — cb')] R 2 ± R(a'c — ac') VF~ S 
y _ __ _« 5 



, _ [a(a'c -ac') + b(b'c - be' )] R 2 ± R (b' a — a'b)\VF t 

where 

F s = (ac - ac) 2 + (be' — b'c) 2 + (ab' - a'b) 2 - R 2 (a 2 + b 2 + c 2 ), 
and 

F\ = (ac - a'c) 2 + (be' - b'c) 2 + (a'b - ab') 2 . 

It is observed that F t and F' s are symmetric in a, b, c, a', 6', e ; that 
x , y', z are finite (for finite a, b, etc.) unless a :b : c = a' :b' : c ; and that 
x', y', z are real or imaginary according as F s = , or F s < . 

The tangent plane to S at (x y' z ) is 

or since 

(II") xx + yy' + i8»' = R 2 . 

This plane contains the tangent lines of all great circles of S, passing through 
(x'y'z). To prove a circle orthogonal to S, itjis only necessary, therefore, 
to prove the tangent to the circle at the point (x y' z ) perpendicular to this 
tangent plane at (x'y'z). The plane is real only>hen F t = . The tangent 
to the circle of intersection of S' and S" at (x'y'z) is the intersection of plane 



O C. S. FORBES : THE GEOMETRY OF CIRCLES 

II and the tangent plane of one of the spheres, say /S', at the point. This 
tangent plane is 

or 

(*' + o')(x - as') + (2/' + b')(y - y') + (»' + c')(« - »') = 0. 

We have the relations 

/2 , ,2 . ,2 D , 

a'aj'-f b'y' -\- c'z'= — P°' . 
Hence the above reduces to 

(as'+ a')a + (y'+ 6')y + (z'+c')z = 0. 

Since both this plane and plane II pass through the origin and (x'y'z), their 
line of intersection does the same, and is perpendicular to the plane II" , tangent 
to S. But this line is the tangent of the circle (S', 8") at the point of inter- 
section with S. Hence circle (#', S") is orthogonal to every great circle of S 
passing through (x'y'z') and is orthogonal to S. At no point in the argument 
have we assumed that (x'y'z) was a real point. Hence Theorem X holds for 
both the real and imaginary cases. 

Theorem XI. Every circle orthogonal to a sphere at one point of inter- 
section is orthogonal at the other also. 

This follows at once from the fact that the circle lies in a plane passing 
through the center of the sphere (Theorem VI), and from the symmetry of the 
construction. 

Theorem XII. There are 00 4 circles orthogonal to a given sphere. 

Theorem V imposes two conditions upon the six parameters of the circle, 
leaving four degrees of freedom. Again there are 00 2 points on the sphere. 
Through each point P and the center of the sphere, 00 planes may be passed. 
In each of these planes 00 circles may be drawn through P , orthogonal to the 
intersection of the plane and the sphere, and hence orthogonal to the sphere 
(Definition III ; also see proof of Theorem X). Therefore through any point 
on the sphere, 00 2 circles may be passed orthogonal to the sphere. Hence, since 
each circle cuts the sphere in a finite number (2) of points, there are 



circles orthogonal to a given sphere. 

Theorem XIII. Two points on the sphere fix one of the orthogonal circles. 
Pass a plane through the points P and P', and the centre of the sphere. In 



ORTHOGONAL TO A GIVEN SPHERE 7 

this plane one and but one circle may be passed through P and P', and orthog- 
onal to the circle formed by the intersection of the plane and the sphere. 

Theorem XIV. All those circles orthogonal to the same sphere, which have 
one point in common, have a second point in common related to the first point 
by inversion with respect to the sphere. 

Let the circles C and C" be orthogonal to a sphere S, and intersect in a 
point P . P may be taken without the sphere. Let S' be any sphere passing 
through C but not through C". Spheres S and S' are orthogonal (Theorem 
VII). Circle O" lies in a plane II drawn through P and O, the centre of S 
(Theorem VI). This plane cuts out of sphere $' a circle C" orthogonal to 
sphere S (Theorem XI ; Lemma, Theorem X). Draw OP . Plane II cuts out 
of IS a circle C . C" and &'" therefore lie in the same plane II with circle C 
and are orthogonal to it. C" cuts OP a second time at B. Therefore by the 
theory of orthogonal pencils of circles * C" also passes through B . That is P 
and B are reciprocal poles with respect to the circle C '. Circle C" lies in the 
same plane with OP and therefore cuts OP a second time. But C lies on 
sphere S' and therefore must cut OP in the point in which OP cuts S' . . This 
point is B since C" lies on S' and passes through B . Hence C and C" inter- 
sect a second time in B. 

We next establish that the analytical relation between the points P and B 
is that of inversion with respect to the sphere S. Assume 0, the center of S, 
as the origin. The equation of £ is 

tf + y* + !» = &. 

Since the configuration is independent of a rotation of axes, the plane II may be 
assumed as 

3=0. 

The circle C cut from S by II is 

x 2 +y 2 =B 2 . 

Let OP be the axis of X and P be the point ( K ) . Since the circle 
C" cut from S' by II , is orthogonal to C, its equation is of the form 

x* + y 2 - 2ax - 2by = - fi 2 . 

Since C" passes through the point ( K ) , 

K 2 -2aK=-P 2 , 
i. e.. 

K 2 + P 2 



Kleix, Einltitung in die hohere Geometric, vol. I, p. 87. 



O C. S. FORBES: THE GEOMETRY OF CIRCLES 

The equation of C" is therefore 

77-2 i r>2 

x 2 ^- x + f-2by=-R\ 

This circle cuts the axis of X in the points (P and B) 

(KO 0) and (R 2 /K0 0). 

Since the coordinates of B do not involve the coefficient b of the equation 
denning C", it is seen that P and B are uniquely related, i. e., since any two 
circles orthogonal to S and intersecting in P , also intersect in B, all such 
circles passing through P will also pass through B. The coordinates of P 
and B satisfy the formulae of inversion 

x'R 2 „ v'R 2 



x + y 



and are therefore reciprocal poles with respect to the circle C 

Since the configuration is independent of a rotation of axes, the relation 
between P and B is unaltered by a rotation of plane II about OP as axis. P 
and B are therefore reciprocal poles with respect to the sphere, and their 
coordinates satisfy the formulae of inversion 

x'R 2 „ y'R 2 „ z'R 2 



x + y' + z' x' + y' + z x" + y + z 

For it has been shown that if OP = K, then OB = R 2 jK. That is 
OP x OB = R 2 . Now let P be the point (x'y'z ) . The point (x"y"z") lies 
upon OP . For the equation of the line OP is 

xy'z = yx'z = zx'y' . 

Substituting the values of the coordinates of (x"y"z"), we have the identity 

x'y'z'R 2 x'y'z'R 2 x'y'z'R 2 

,2 , ~i%~ \ ~i% = /2 , J2~; 72 == ,2 , ~i2~\ 72 * 

x + y + z x + y + z x + y + z 
Furthermore the distance 



OP=V x ' 2 + y' 2 + z 
and if (x"y"z") is denoted by B, 



V(x' 2 + y' 2 + z")R 2 

UB = 72— ; 72— — 72 

x +y +z 

.-. OPX OB = R 2 . 



ORTHOGONAL TO A GIVEN SPHERE y 

Hence to every point without a sphere S there corresponds a point within. 
These two points are related by the formulae of inversion. If two circles orthog- 
onal to the sphere S pass through a same point without the sphere, they pass 
through the corresponding point within. Two circles orthogonal to a same 
sphere, therefore intersect in two points or not at all. No two such circles can 
intersect in two points without or within the sphere, else they intersect in two 
other points corresponding to these two, and therefore coincide. It is readily 
seen that the relationship between corresponding points is reciprocal. In other 
words throughout the argument the terms point without and point within may 
be interchanged. 

Particular attention is called to Theorems X, XII and XIV, which are of 
especial consequence in the subsequent chapters. We repeat these theorems for 
the convenience of the reader. 

Theorem X. The intersection of two spheres orthogonal to a given sphere 
is a circle orthogonal to that sphere. 

Theorem XII. There are oo 4 circles orthogonal to a given sphere. 

Theorem XIV. All those circles orthogonal to a same sphere, which inter- 
sect in one point, intersect in a second point the reciprocal pole of the first 
point with respect to the sphere. 

It is noted in passing that Theorem X is but a generalization of the theorem : 
The intersection of two planes perendicular to a third plane is a line perpen- 
dicidar to the third plane. 

We pass from these preliminary investigations to the main body of our paper, 
and shall henceforth make exclusive use of systems of circle-coordinates derived 
from the Darboux pentaspherical coordinates. 

CHAPTER II 

Coordinates of the Circle 
A. Pentaspherical Coordinates 
1. Choose five mutually orthogonal spheres 

S x , S 2 , S s , S t , s & . 
Let their radii be 

i? 1? i? 2 , i? 3 , i? 4 , i? 5 . 

Any point is determined by five Darboux * pentaspherical coordinates, 

ry. rt. ry* rp rf> . 

Jj x , Jj 2 , Js 3 , ^/ 4 , ^ 5 , 

where 

P. 
x i = Pjf (i=l,2,---,5); 



Daebodx, Theorie des Surfaces, vol. 1, book II, chap. 



10 C. S. FORBES : THE GEOMETRY OF CIRCLES 

P. denoting the power of the point with respect to the sphere S { , and p being 
an arbitrary factor. The five coordinates are homogeneous, and are connected 
by the identity 

(1) x* + x$ + xl + xl + xl = 0. 

Any equation of the form 

5 

(2) a x x x + a 2 x 2 -f a 3 x 3 + a 4 x 4 + a & x b = ^ a t x t = , 

i 

where the quantities a. are constants, represents a sphere. * The equations of 
the five fundamental spheres of reference, S it are 

(3) as, = 0, * 2 =0, x 3 = 0, x 4 =0, cc 5 = 0. 
These spheres intersect in ten circles 

(4) ^ Xi =X. = Q (i,j = l,..-,5;i*j); 
and in ten pairs of points 

(5) x i = x J = x k =0 {i,j f k = l,-'-;6;i+j + k). 

2. The condition that two spheres 

5 5 

(6) 2>.cc. = 0, E 6 ^.^ 
shall be orthogonal is 

(7) a 1 b 1 + a 2 b 2 + a 3 b 3 +a i b i +a & b 5 =0. 

3. We are considering the geometry of circles orthogonal to a given sphere. 
The choice of the five fundamental spheres, S i , allows that the given sphere be 
taken as any one of the five. We choose to take it as S s , and to give it a real 
positive f radius. Although this latter restriction is of no advantage algebra- 
ically, it serves to clarify the geometrical conceptions. 

4. The equation of a sphere orthogonal to S., is of the form 

(8) a Y x x -h a 2 x 2 -f a 3 x 3 + a 4 x 4 = J2 a i x i = ® • 
[See (3) and (7).] 

There are oo 3 such spheres, since (8) contains three parameters (Cf. Theorem 
IV). Any two such spheres intersect in a circle, real or imaginary, orthogonal 
to S 5 . This has been proved in Theorem X for both the case where the circle 
of intersection is real, and where it is imaginary. There are oo 4 circles orthog- 
onal to S 5 (Theorem XII). 

*.Cf. Koenigs, Contributions d, la theorie du cercle dans Vespace, Ann ales ••• Toulouse, 
1888, p. 1 ; Daeboux, op. cit. 

fOne of the five spheres must have a negative radius. See Klein, Einleitung, vol. 1, p. 102. 



ORTHOGONAL TO A GIVEN SPHERE 11 

Although there are six parameters involved in the equations of the two 
spheres intersecting to form the circle, the latter can be thus generated in oo 2 
ways. All the circles orthogonal to S 5 may be generated by the intersections 
of the spheres orthogonal to S 5 . 

5. Since we treat exclusively of the spheres and circles orthogonal to S 5 , we 
shall use the terms sphere and circle to mean respectively, sphere orthogonal to 
S 5 , and circle orthogonal to S 5 . 

The configuration of reference consists therefore of the four mutually orthog- 
onal spheres 

S v S 2 , S 3 , S t , i.e., a;.= (i=l,---4). 

These spheres intersect in six circles 

and in four pairs of points, 

»< = », = ** = (<,i,* = l | ...4;*+i + *). 

Attention is called to the similarity between this tetrasphere of reference and 
the tetrahedron of reference in line geometry. The following notions correspond. 



Teteahkdron 


Tbteaspheee 


4 faces 
6 edges 
4 vertices 


planes 

lines 

points 


spheres 
circles 
pairs of points 



It is noted that the four spheres of reference belong to the assemblage of 
spheres orthogonal to S 5 , but that JS. itself does not. The coordinates of a 
point referred to this configuration are tetraspherical. 

B. Point-pairs 

6. Three spheres of the assemblage intersect in two points, one within and one 
without S s (Theorem XIV). All those circles (or spheres) which pass through 
one of the points pass through the other also. Conversely, if two circles of the 
assemblage intersect in two points, these points are related by inversion. We 
denote two points so related as a point-pair, or simply as a point. 

The pentaspherical coordinates of the point-pair may arbitrarily be taken as 
those of the outer element, but we now show that inasmuch as a point is fixed by 
the ratios x 1 :x 2 :x 3 :x i , it makes no difference whether the inner or outer element 
is selected. 

Let (x'y'z') and (x"y'V) be the elements of a point-pair. We have by the 
formulae of inversion with respect to the fundamental sphere 



12 C. S. FORBES: THE GEOMETRY OP CIRCLES 

(S 5 ) x 2 + y 2 + z 2 = R 2 , 

the relations 

"_ xR2 '"_ y' R2 "_ z ' R2 

X ,2 , ,2 , ,21 V — ,2 , ,2 , ,21 Z — ,2 , 72~~ J* ■ 

x + y + z x + y + z a3 + y +S ; 

Let S. , one of the fundamental spheres, not S s , be 

x 2 + y 2 + z 2 + 2ax + 2by + 2cz + R 2 = . 

S. is orthogonal to S s . 

With reference to (x'y V) 

P. x' 2 + y' 2 + z' 2 + 2ax' + 2by' + 2cz' + R 2 

*i = PR=P R { 

( t = 1, • • • , 4 ; p arbitrary) . 
With reference to ( x"y"z" ) 

(x' 2 + y' 2 + z' 2 ) 2ax' + 2by' + 2cz 

, ,2 , ,2 , ,2»2 Xt T ,2 , ,2 , ,2 JX ~T -"> 

(x + y + z ) x + y + z 
*-=P V ' ^ 

P 2 x' 2 + y' 2 + z' 2 + 2ax' + 2by' + 2cz' + R 2 



Hence we have the relations 

^ 2 

(9) «i = -72- ,2 , ,2 X i (» = l,---,4). 

x +y +z 

7-> ,2 , ,2 , ,2 t->2 

P, x + y + 2 — R 
X * = Pr=P R • 

,2 , ,2 , ,2 

• x +y +z t?i m 

, ,2 , ,2 . ,2,2 SX — -* 1 t-,2 ,2 , ,2 , ,2 ,,2 

(x + y + a ) P 2 x +y +z — R 

x s = P p = -p-*-, — ,2 , ,2 5 ' 

-K x + y + z -°- 

i. e., 

P 2 

( 10 ) < = ~ ,2 , ,2 , ,2 *; • 

x + y + z 

The theorem follows : 

The pentaspherical coordinates of the elements of a point-pair are re- 
spectively proportional, save that the coordinates x. differ in sign, i. e., 

The coordinate x 5 of the inner point is always negative.* For if the inner 
point is (abc) 

P 5 = a 2 +6 2 + c 2 -P 2 <0, .-. x 5 = p P ^<0. 



For p positive. 



ORTHOGONAL TO A GIVEN SPHERE 



13 



Therefore to every point-pair there is a definite set of ratios x 1 :x 2 :x 3 :x i , 
and conversely, to every set of these ratios, there is a definite point-pair. The 

5 

coordinate x 5 may be determined from the identity ^x\ = 0, the sign being ± 
according as the { ?£% r } element of the point-pair is taken. 

Correspondence I. To every point without a sphere S 5 corresponds a 
unique point within, and reciprocally. 

To the center of the sphere, however, corresponds the plane at infinity and 
reciprocally. If the radius of the sphere is zero, the sphere is a point which cor- 
responds to every point of space. 

4 

C. Dual Interpretation of Y,a i x i = 
i 

4 

7. The equation '£a i x i = , may he interpreted in a dual sense. If we regard 

the quantities a { as constant, and the quantities x t as variable, equation (8) 
represents a sphere as the locus of its points. By virtue of the identity (1), a point 
is known when four of its coordinates are given. (Cf. Section 6.) Three points 



determine the sphere, since they determine the ratios of the quantities a i uniquely, 
provided the points are not on a same circle, i. e., 



(11) 



+ 0, 



The same sphere is equally well determined by any three points, whose coordi- 
nates are of the form 

(12) X i = \x' i +\x" i -\-\x"f 0' = 1, •■•, 4; \, \, ^arbitrary). 

Conversely every point whose coordinates are of the form (12) is on the sphere.* 
Reciprocally if the quantities x i are regarded as fixed, equation (8) repre- 
sents a point-pair x. as the envelope of the spheres which pass through it. 
Three sets of sphere coordinates 



fix the point, provided the three spheres do not pass through a same circle, i. e. 
a\ a' a' a. 



(13) 



+ 0, 



* Cf . Keysee, Plane Geometry of the Point in Space of Four Dimensions, pp. 311-313 ; Kobnigs, 
La Geometrie Beglee, Annales-Toulouse, 1889, p. 11, et al. 



14 



C. S. FORBES: THE GEOMETRY OF CIRCLES 



The point is equally well determined by any three spheres whose coordinates 

are of the form 

(14) a i = \a'. + \a'l + \a"! (t = l, •■•4; \, \, A 3 arbitrary). 

Conversely every sphere whose coordinates are of the form (14), passes through 
the point. 

D. Coordinates of Circle as Envelope of Spheres 

8. All the spheres passing through a given circle constitute a pencil of 
spheres. If the circle is given by the intersection of the spheres 

Eax = 0, £&.£c.= 

the coordinates of the spheres of the pencil are of the form 

a. = \ a. + \ b. ( Aj , \ arbitrary) . 

The circle is equally well defined by any two spheres of the pencil 

'£(a i +\b i )x i = Q {x = \i\). 

9. In particular it is defined by any two of the four spheres of the pencil, 
which are orthogonal, each to one of the four fundamental spheres 

x x = , x 2 = , x % = , x t = . 

The equations of these four spheres of the pencil are 

* +Pl2 X 2 + PlS X S+Pu X t = ' 
P 2l X x + * +P 2 3 X 3+P2i X i = > 
Psi X l+Ps2 X 2 + * +P& X l= > 

[p il x 1 +p i2 x 2 +p i3 x 3 + * =0, 

a.b. — a.b. (»,* = !, ••■, 4). 



(15) • 



where 
Obviously 



P« : 

Pa = ° ! 



P*= -Puc 



Expanding the determinant 



a i a 2 a s a i 

\ h h \ 

<h <*2 a 3 a i 

\ h h K 



• Cf. Koenigs, La Oeometrie Beglee, Annales • • • Toulouse, 1889, p. 6. 



ORTHOGONAL TO A GIVEN SPHERE 15 

in terms of its second order minors, we obtain 

(16) *> O ) = PuPu + PuPv + PuPx = ° • 

Although two of equations (15) are sufficient to determine the circle, all four 
are retained for the sake of symmetry, and the six * coordinates p ik are taken 
as the homogeneous coordinates of a circle orthogonal to the fundamental sphere 
S 5 . The identity (16) reduces the degree of freedom of the system to four as 
is required by Theorem XII. 

E. Coordinates of Circle as Locus of Point-Pairs 

10. Reciprocally two point-pairs 

^ J a i x i = 0, H«i2/i = 

determine a circle, and the same circle is determined by any two points of the 
range 

]T ( x i -f Xy . ) a. = (a arbitrary ) . 

(By range of points is meant all the point-pairs on a circle which contains the 
generators of the range.) In particular, the circle is determined by any two of 
the four points of the range lying on the fundamental spheres. The equations 
of these points are 

* + ?12«2 + ?13«3+?14«4 =0 > 
?21«1+ * +2'23«3 + ?24«4= > 
?3i a i + ?32«2+ * + ?34 O '4 = = > 

q 41 a 1 + q 42 a 2 + 4s a 3+ * =0, 
where 

?,* = va-**^ (t,&=i,---,4). 

?vi = °> ?«=-&• 

The identity exists 

( 18 ) G, (?) = ?12^34+? 1 3?42 + ^?23= - 

11. As shown in the precisely similar algebra of line geometry,! if p ik and q. k 
are the coordinates of a same circle, 

/1Q\ Pj2 Pi* Pli Pji .^42 P23 

<lu~ ?42 _ fe" 2l2 _ ?13~ ?U 



(17) 



* Cf . Pluckee, Neue Geornetrie des Raurnes, p. 2 ; Cayley, Six Coordinates of a Line, Collected 
T orhs, vol. 7, p. 66. 
f Pluckee, Neue Geornetrie, p. 5. Koeniss, La Geornetrie Beglee, Annales---Toulouse; 



16 C. S. FORBES : THE GEOMETRY OF CIRCLES 

Hence six quantities <y ih , such that 

7i2 = PPa = °<lu ' 7i3 = PPn = °V42 > etc - ( P , ^ arbitrary ) , 

may be assumed as the homogeneous coordinates of a circle, without regard 
to the method of generation, or if the coordinates are taken in a certain order 
they give the circle as an envelope of spheres, or if the same coordinates 
are taken in a different order the circle is regarded as the locus of its points 
(i. e., point-pairs). Conversely, given six quantities y ik , which satisfy the identity 

O)( 7 ) = 0, 

they may be taken as the coordinates of a circle, and be interpreted in a dual 
sense according to the order in which they are taken. 

F. Conditions that Two Circles may Intersect 

12. Two circles intersect* when their generating spheres (points) have a point 
(sphere) in common. Let the given circles be 



(20) 

(21) 
T 

(22) 



* + 7l2» 2 + 7i 3 *3 + Yu X 4 = ' 
721*1+ * + 723*3 + 7 M aj 4=°i 

* + 712*2 + 7i 3 *3 + Vu X * = ' 
7^ «, + * + 7 23 *3 + 724*4 = ° • 

These four spheres (points) have a point (sphere) in common, provided 

7i2 7i 3 7 U 

7 21 723 7 24 ^^ 

7i 2 7; 3 7i4 

721 7 2 3 7 2 4 

7« = 7m 
®(7) = 



By the aid of 

and 

this reduces to 

(23) fi>( 7 , 7') = 7 12 7^ + 713^2 + 7l47 23 + 7^2 734+ 7i 3 742 +7^723 = °- 



This is therefore the condition that the circles 7^ and <y' iJe shall intersect in 
a point-pair, or dually shall lie upon a same sphere. 



t Cf. Koenigs, La Geometrie JSeglee, op. cit., p. 9. 



ORTHOGONAL TO A GIVEN SPHERE 



17 



G. Conditions that a Given Circle may lie on a Given Sphere or pass 

through a Given Point-pair 
13. We seek the condition that a given circle (envelope of spheres) may lie 
upon a given sphere. Let the sphere be given by 

(24) c x x x + c 2 x 2 + C3CC3 + c i x i = ; 

and the circle by 

a x x x + a 2 x 2 + a 3 x 3 + a 4 as 4 = , 

b x x + b 2 x 2 + b 3 x 3 + b i x i = . 

The condition is that the sphere is one of the spheres of the pencil of spheres, 
which is defined by (25), i. e., 

c. = \a i + \ b i ( \ , \ arbitrary ) , 

This involves the two conditions 



(25) 



(26) 



a x b x c x 

« 2 h C 2 

«3 6 3 C 3 

a A l A c A 



14. Reciprocally, the condition that a circle (locus of points) given by two 
points shall pass through a third point, is that the point shall belong to the 
range defined by the two points which give the circle. 

Let the point be 

(27) a x z x + a 2 z 2 + a 3 z 3 + a^ = , 

and the circle be 

J a x x x + a 2 x 2 + a 3 x 3 + a^ = , 

I «i2/i + a 2 y 2 + « s 2/ 3 + ^Vi = ° • 
The condition is 

z.= \x i + \y i 
This involves the two conditions 



(28) 



(\, \ arbitrary] 



(29) 



x x y x z x 

X 2 Vl Z 2 

x 3 3/3 h 

*l 2/4 z i 

15. The condition that a given circle locus of points shall lie on a given 
sphere, is that both points which define the circle shall lie on the sphere. If the 
sphere be 

a x x x -f a 2 x 2 + a 3 x 5 + a^x^ = , 



18 C. S. FORBES: THE GEOMETRY OF CIRCLES 

and the points be y i and z., the conditions are 

Kyi + « 2 y 2 + « 3 y 3 + ^y* = ° > 

(30) { 

[a l z l -f a 2 z 2 + a 3 z s + a i z i = 0. 

16. Reciprocally, the condition that a given circle, envelope of spheres shall 
pass through a given point, is that both spheres which define the circle shall pass 
through the point. If the point is 

a x x x + a 2 x 2 + a 3 x 3 -f a 4 a; 4 = , 

and the circle is defined by the spheres b t and c i , the conditions are 

[&!«! + b 2 x 2 + b 3 x 3 + b 4 x t = , 

( 31 ) 1 

[ c 1 cc 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 = . 

17. The last four sections exhibit the similarity between the reciprocal alge- 
bras of the circle regarded as an envelope of spheres and as a locus of points. The 
necessary and sufficient conditions for the union of the circle and sphere, and 
the circle and point in the one theory, are algebraically identical with the neces- 
sary and sufficient conditions for the union of the circle and point, and the circle 
and sphere in the reciprocal theory. 

CHAPTEE III 
The Linear Complex of Circles 

18. Equations of relation between the coordinates of the circle serve to select 
special systems of circles from the oo 4 circles of the space. The system defined 
by a single equation is called a complex. If the equation be linear, the complex 
defined by it is called linear. 

A. Linear Complex of Circles Regarded as Envelopes of Spheres 

19. The equation of a linear complex of circles regarded as envelopes of 
spheres may be written * 

( 32 ) A uPl2 + A l S Pl3 + AiPu + A 2zP 23 + A 4zP& + A uP& = • 

Expanding, we obtain 

A2OA -«A) + AsKA - a A) 

(33) + A^a^-aJJ + A 23 (a 2 b 3 - a 3 b 2 ) 

+ A2OA ~ <*A) + 4«K 6 4 ~ °A) = - 

* Cf . Pluckee, Neue Geometrie, p. 27. 



ORTHOGONAL TO A GIVEN SPHERE 19 

Rearranging according to terms in a. , we obtain 

(+AA + As 6 . + AA)«i 

+ (-A 12 b 1 + A 2S b s -A i2 b i )a 2 
(34) 

+ (- AA- AA + AAK 
+ ( - AA + A a\ - AAK = °- 

If the quantities b { are fixed, they define a sphere 

(35) E&,a;.= 0. 

If we regard the quantities a i as variable, equation (34) is the equation of a point, 
the envelope of all the spheres of the circles of the complex, lying upon the 
sphere (35) ; in other words all circles of the complex lying upon the sphere (35) 
pass through the point (34). Hence, to each sphere of space corresponds with 
respect to each complex one point. The point lies upon the sphere since 

(36) a. = b. 

satisfies equation (34). Therefore 

All the circles of the complex lying on a sphere pass through a point of the 
sphere and form a pencil of circles. 

B. Complex of Circles Regarded as Loci of Point-Pairs 
20. Reciprocally, the equation of the complex may be written 

( 37 ) Al?M + As 242 + A-l&S + Asffl* + A 2 ?13 + A u1l2 = ' 

[See (19).] 
Expanding, we obtain 

A«( a i.y« - x iVz) + AsC 35 ^ - **Va) 

(38) + A u (x 2 y 3 - x 3 y 2 ) + A 2Z {x lVi - x iVl ) 

+ AsC^s - x sVi) + A 3i( x iy 2 ~ x *Vi) = ° • 
Rearranging according to terms in x { , we obtain 

( + Aa^ + A^s + A M*i 

• J r{- A izV i + A u yz- A uVi) x 2 

(39) 

+ ( + A 2 ^ - A*y 2 - A a Vi) x z 
+ (-A 2 y 3 + A 3 y 2 - A32/iK= °- 



20 C. S. FORBES: THE GEOMETRY OF CIRCLES 

If the quantities y i are fixed they determine a point 

(40) £&^=0. 

Then, if we let the quantities x. vary, equation (39) represents a sphere, the locus 
of the points of the circles of the complex, which go through the point (40), in 
other words, the locus of the circles themselves. The point (40) is on the sphere 
(39), since 

(41) x { = y. 
satisfies (39). Therefore 

To each point of space corresponds, with respect to each complex, a sphere 
passing through it, and all the circles of the complex passing through the 
point lie on the 



C. Polar Spheres and Pole Point-Pairs 

21. We denote a point and a sphere related as in the last two sections as 
pole and polar with respect to the complex. 

Correspondence II. Every complex correlates the point-pairs and spheres 
of the space, so that, toith respect to the complex, to every point-pair there cor- 
responds a sphere polar to the point-pair, which contains all the circles of the 
complex which pass through the point-pair, and to every sphere corresponds a 
point-pair, pole of the sphere, through which pass all the circles of the complex 
lying upon the sphere. 

Correspondence III. A triple correspondence is set up by each complex 
between the point-pairs and the spheres of the space and the circle pencils of 
the complex, such that a point-pair, a sphere and a circle pencil correspond 
uniquely and are united in position. 

The pole of the sphere 
(42) Hb.x i =0 

is, from (34), the point whose coordinates are 

y* = - A u h i + A zA - A 2 6 4> 

■?/, = — A, b, 



(43) 



y 4 = - A Xi \ + A i2 b 2 - A u b z . 
The polar of the point y i is the sphere whose coordinates are 

b i= AzVi + ^yz + ^y^ 
\=- As?/* + A uys - A UV^ 



(44) 



[See (3 9).] 

A22/4- A uy* - A 4 2 y^ 
. \ = - A ™y* + A uy 2 - A 2,y, • 



ORTHOGONAL TO A GIVEN SPHERE 21 

Let 

(45) co(A) = A 12 A 3i + A n A i2 + A u A 2S . 

If the last three of equations (43) be multiplied respectively by A 3i , A i2 and 
A^, and the results be added, we find 

- <o(A)\ = A u y 2 + A i2 y 3 + A 23 y,. 

Similarly we obtain 

-a>(A)l 2 = -A 3i y l + A u y 3 -A l3 y i , 

- v(A)b s = - A i2 y x - A u y 2 + A 12 y 4 , 

- 0i (A)b i =-A 23 y 1 + A ls y 2 -A 12 y 3 . 

Since only the ratios of the quantities b { are of concern, these equations are 
equivalent to (44). Similarly (43) may be gotten from (44). 

D. Second Proof of Fundamental Correlation 

22. The notion of pole and polar with respect to a complex, is so important 
that we give another proof of this fundamental correlation. 
The equation of a linear complex is * 

(46) EA'A=° (i,ft = l > -.-,4,i + A); 

ik 

those of two spheres are 

(47) £^.= 0, Z^=0. 

These two spheres, S and S' , intersect in a circle, whose coordinates are 

The condition that this circle belong to the complex is that its coordinates sat- 
isfy equation (46), i. e., 

ik 

or 

(49) Z(m:aa)=o, 

provided the summation is made under the restrictions 

(50) A ih =-A H ; A ii= 0. 
Consider another sphere /S", 

(51) Z l i x <=°> 
where 



'Cf. Koenigs, Contributions,etc, Annales • • Toulouse, 1888, p. F. 12. 



22 C. S. FORBES : THE GEOMETRY OF CIRCLES 

(52) *<=XA*<V 

k 

The equation [see (49)] 

(53) 5(»«?^«0-2ftl<"-<>. 

expresses that the spheres /S' and £" are orthogonal. Furthermore, from (48) 
and (49), 

(54) £(«,-E4^) = E^ = o, 

i. e., the spheres # and £" are orthogonal. The circle of intersection of S' and 
S, is therefore orthogonal to S" ; but the equation of S" 

£(£^><=o, 

is independent of b . , and in general different from 8. . Hence the theorem : 
All the circles of a linear complex which lie upon a sphere S are orthogonal 
to a second sphere S", which is called the conjugate of S with respect to the 
complex. 

Hence the circles of the complex lying upon S are orthogonal to two spheres 
S" and S 5 . These circles therefore lie in planes, which pass through the cen- 
ters of both spheres (Theorem VI). These planes form an axial pencil, whose 
axis (i. e., the line joining the centers of S" and S 5 ), cuts S in two points. 
The circles of the complex lying upon S all cut the axis twice and therefore 
must pass through these two points. The theorem follows : 

All the circles of the complex which lie on a given sphere pass through a 
point-pair, the pole of the sphere. 

The pole of a sphere 

(55) 5:^^=0, 

may be found as follows : Construct the sphere S" 
(56) ZdX^J^O, 

i k 

and join its center to the center of # 5 . The intersections of this line with (55) 
are the point-pair, pole of (55). 

23. It is seen that, in general, but one of the circles of a pencil of circles 
lying upon a sphere will pass through the pole of the sphere, and belong to the 
complex. We now prove * that if the circles of a complex, whose equation is 

(57) F{ Pik ) = Q, 

*Cf. Pluckeb, Neue Geometrie, p. 18. 



ORTHOGONAL TO A GIVEN SPHERE 23 

are so distributed that but one in general of the circles of a pencil belongs to 
the complex, then F is a linear function. 
Let 

\Pih + \P"ik (\,\ arbitrary ) , 

be the coordinates of the pencil. The condition that but one of the circles of 
the pencil shall belong to F is that the equation 

F ( \ P ik + \P"ik ) = ( f0r SOme \ "• X 2 ) 

shall be of first degree in \ : \ . F is therefore of the form 1A ik p ik — . 

E. Comparison between Circle Geometry and Associate Theories 

24. Analytically the two theories are seen to be identical. Transition from 
one theory to another is effected by the following exchange of notions : 



Assemblage of Circles 


Assemblage of Ltnes 


Circle 
Sphere 
Point-pair 


Line 

Plane 

Point 



Each system is a four-dimensional assemblage. A circle is determined by two 
spheres or two point-pairs, just as a line is determined by two planes or two 
points. The analytical condition that two circles or two lines intersect is the 
same. The polar theory of sphere and point-pair is precisely that of plane and 
point. The oo circles of a complex lying on a sphere constitute a pencil, as do 
the oo lines of a complex lying in a plane. The relationship can be shown more 
clearly however, and an actual correspondence be set up between the lines of 
space and the circles orthogonal to a sphere by means of the double-point geom- 
etry of Cosserat.* He takes as element a pair of points upon the surface of a 
sphere, and as configuration of reference four circles upon the sphere. This 
double-point has six coordinates p ik , and if the four circles of reference be taken 
as the intersection of our four spheres of reference, S t , with S 5 , the coordinates 
p ik of the double-point of Cosserat are precisely the coordinates of the orthog- 
onal circle of S s fixed by the double-point (see Theorem XIII). Furthermore, 
Cosserat shows that if the four centers of the spheres $. be taken as the ver- 
tices of a tetrahedron of reference the coordinates p ik of the line through the 
double-point are, save for a factor of proportionality, the coordinates p ik of the 
double-point. Hence the coordinates of a line and of an orthogonal circle 
through the same double-point differ only by a factor of proportionality. 

* Cosserat, Sur le cercle comme un element generateur de Vespace, Annales ■ • • Toulouse, 
1889, E 40. 



24 C. S. FORBES: THE GEOMETRY OF CIRCLES 

Correspondence IV. To every sphere orthogonal to S 5 corresponds 
uniquely the circle of intersection in dovMe-point geometry, and the 'plane of 
that circle in line geometry. To every circle orthogonal to S. corresponds 
uniquely a double-point upon S 5 , and the line through the double-point. 

Furthermore with respect to any complex the pole of any plane is the point 
where it is cut by the line joining the elements of the point-pair, pole of the 
corresponding sphere with respect to the circle complex whose equation is the 
same; and reciprocally. 

Care must be taken not to confuse point-pair and double-point. The former 
is two points, one within and one without the sphere S 5 , and uniquely related 
by the formulae of inversion with respect to S 5 ; the double-point is a pair of 
points upon the surface of S. , either of which may be again paired with any of 
the oo 2 points of the surface. There are oo 4 double-points and but oo 3 point- 
pairs. 

25. It is of iuterest to note that the plane * geometry of the point in space of 
four dimensions belongs to the same family of reciprocal, four-dimensional geom- 
etries, as does both line geometry and the present theory of orthogonal circles. 
Transition may be effected by the following exchange of notions : 



Assemblage of Cikcles 



Assemblage of Planes 



Circle 
Sphere 
Poiut-paii 



Plane 

Line 

Lineoid 



"We return to the development of circle geometry, bidding the reader keep in 
mind the associate theories to which we have called attention. 



CHAPTER IV 

Conjugate Circles, Pole Point-pairs and Polar Spheres 

A. General Theorems 

26. If a point and a sphere are united in position, their respective polar 
and pole are united in position.j- 
For let x\ and d[ be respectively a point and a sphere, such that 



(58) x»; 



*EJEYSEB, Plane Geometry, etc., op. cit. 

fCf. Koen'IGS, La Geometrie Eeglee, chap. 2. Mobius, Vber Figuren im Baume, Crelle's 
Journal, vol. 10, p. 321. 



ORTHOGONAL TO A GIVEN SPHERE 25 

Let a\ and x" be the polar and pole respectively, of x\ and a" with respect to a 
complex C. To prove 

The circle of intersection of the spheres a\ and a", being contained in a', and 
containing x\ (58), the pole of a\, belongs to the complex. Therefore since it lies 
upon a", it contains x".. Therefore 

E« = 0. 

27 . Let p ik be any circle. Every generating sphere S' of p ik is united in 
position with every generating point P of p. k . Hence every pole P' of the 
spheres S\ is united in position with every polar S of the points P. That 
is, the points, P', and the spheres, S, generate one and the same circle p' ik . 
Two circles, p ik andp^, so related that each of them is the locus (or envelope) 
of the poles (or polars) of the generating spheres (or points) of the other, are 
called conjugate circles. 

28. The following theorems * result at once by analogy with the line geometry 
of ordinary space, and the plane geometry of the point in four-space. 

Every circle of a complex is self-conjugate with respect to that complex. 

Every self-conjugate circle belongs to the complex. 

Two distinct conjugate circles cannot pass through a same point, else they 
would belong to the complex and be self conjugate. 

Two distinct conjugate circles cannot lie on a same sphere. 

If two conjugate circles have each a sphere (point) in common with a third 
circle, this third circle belongs to the complex. 

: complex { pai Kesc 
conjugate circles, it ^J^tZu^} <* *™ \ P oint- P airS 

All the circles intersecting two conjugate circles belong to the complex. 

29. If two circles intersect, their conjugates also intersect. 

For let p and p" be two circles intersecting in the point P. The conjugates 
of p andp" lie on a sphere JS, polar of P, and therefore intersect. 

Correspondence V. To the oo 2 circles of a point-pair correspond the oo 2 
circles of a sphere polar to that point-pair with respect to a complex, and 
reciprocally. 

B. Special Complexes of Circles 

A complex is called special when all its circles intersect a same circle, the 
directrix of the complex. The condition that two circles p ik and p' ik may inter- 
sect is 

( 59 ) PnPu + PisPu +••• + ^34X2= °- 

[See (22).] 

*Cf. Koenigs, La Geomelrie Beglee, chap. 1. Keysee, op. eit., p. 117. Mobius, op. cit., etc. 



26 C. S. FORBES: THE GEOMETRY OE CIRCLES 

If the quantities p' ik are fixed this is the equation of a complex, all of whose 
circles intersect p\ k , i. e., a special complex whose directrix is p' ilc . The con- 
ditions that the complex 

may be special is therefore 

(60) K^*) = °- 

32. Every pair of complexes,* C and C , of which one C is special, deter- 
mines another special complex C" such that the assemblage of circles common 
to C and C", is identical with that common to C and C. 

The director circles of C" and C" are conjugates with respect to C. C is 
one of the pencil of complexes determined by 

X x C -f- X 2 C " ( \, \ arbitrary ). 

From this may be derived the coordinates of a circle p" ik conjugate top^. 

(61) ^—^-zkk^ 

Also given a complex C and a circle p ik , the theorem follows : 
p. k is self -conjugate or not, according as it belongs, or does not belong to C ; 
if p' ik and p" ik be any two circles, their conjugates are distinct, or not, accord- 
ing as C is non-special, or special. In case C is special the directrix is con- 
jugate to all circles, itself included. 

33. The following theoremsf result at once from the definitions of pole and 
polar. 

The polars of the points of a sphere pass through a common point, the pole 
of the sphere. 

The poles of the spheres of a point lie upon a same sphere, the polar of the 
point. 

The conjugates of the circles lying upon a sphere all pass through the pole 
of the sphere, and the conjugates of the circles passing through a point, all 
lie upon a sphere, the polar of the point. A circle passing through two points 
has for its conjugate the circle formed by the intersection of the spheres, polar 
to these points. 

The circle formed by the intersection of two spheres, has for its conjugate 
the circle passing through the poles of the spheres. 

C. Coordinates of Conjugate Circles 

34. We derive the coordinates of a circle conjugate to a given circle, by a 
method slightly different from that used in section 32, and based directly upon 
the principles of section 33. Given a complex 

* Cf. Keysek, Plane Geometry, etc., pp. 318-320. Koenigs, La Geometrie Mglee, op. cit., p. 24. 
f Cf. MOBIUS, fiber Figuren im Baume, pp. 329-336. 



ORTHOGONAL TO A GIVEN SPHERE 



27 



(62) 



and a circle p ik defined by the spheres, S and S' , 

Px 2 x 2 + Pi 3 x s + Pu x i=°, 

P2l X l+P23 X 3 + P2i X 4=°i 

we seek to find the coordinates of p' Uc conjugate of p ik . The conjugate of p ik 
is the locus of the poles of the generating spheres of p ik , and is fixed by any two 
of these poles, particularly by the poles of (62) themselves. 
These poles are [see (43)], 

[x[= A 12 p 12 + A lsPl3 + A u p u , 

X 2= A 2 3 Pl3~ A 4 2 Pu> 

X 3 A 23 Pl 2 + A 3i Pw 

.<= A 4 2 Pl2- A SiPl3' 



(63) 
and 
(64) 



X " 2 = ~ A l 2 P 2 l + 4^ - ^42^24' 
X l = ~ A l S P2l + A UP2^ 
X 'i=- A uP 2 l- A 3 i P 2S ' 

A circle passing through these two points is fixed by any two spheres passing 
through the points, in particular by the spheres passing through these points 
and one of the points 

(65) x 2 = x 3 = x i == 0, x x = x 3 = cc 4 = , 

i. e., one of the vertices of the configuration of reference [see (5)] . The equa- 
tions of these two spheres are 



(66) 



and 



(67) 



x l 


X 2 


X 3 


x, 


x 'l 


x' 2 


X '-6 


x 'i 


x" 


x'i 


x'l 


x'l 


1 











x l 


x 2 


x 3 


x i 


x[ 


X 2 


x' 3 


»; 


x'l 


x'l 


x'i 


x'l 





1 









28 C. S. FORBES: THE GEOMETRY OF CIRCLES 

Expanding we obtain 

\a 3i x 2 + a i2 x 3 + a 23Xi =0, 
(68) i 

where 

a i!c ~ X i X k ~ X k X i' 

Since the spheres (68) are orthogonal to the fundamental spheres, whose equa- 
tions are 

Xl = , x 2 = , 

the quantities a ih may be taken as the coordinates of the circle (see section 9). 
That is 

Pl2 = a n = X 3 X i 





Pl3 ~ a 42 = X i X 2 X 2 X i ' 


(69) 


Pl4 = a 23 == X 2 X 3 X Z X 2 ' 




p 23 = a u = x 1 x i x i x l , 




p i2 = a 13 = x x x $ x 3 x 1 . 


From the form of these equations and the condition 


»(!>) = 0, 
it follows that 




Pu — a \2 — X \ X 2~ X 2 X l ' 



But these are the coordinates q ik of the circle regarded as a locus defined by the 
two points x\ and x". . In fact, the last theorem of section 33 states: That the 
conjugate of a circle formed by the intersection of two spheres is the circle 
fixed by the poles of these spheres. 

The theorem follows : 

If a circle is given by coordinates p ilc and its conjugate by coordinates q ik , 
the spheres which define the coordinates p iJc are the polars of the points which 
define the coordinates q ik , and reciprocally. 

It is found on trial that the results of (61) and (69) agree save for a factor 
of proportionality. 

D. Construction of the Complex 

35. Let the symbols (P, P') and (S, S') denote the circles drawn through 
the points P and P', and lying on the spheres S and iS' respectively. Let the 
symbol ( P', P", P'" ) denote the sphere drawn through the three points P', P', 
P" and (/S", S" , S'") denote the point of intersection of the spheres S', S" 
and &'".. 



ORTHOGONAL TO A GIVEN SPHERE 



29 



36. Five non-intersecting circles of a complex being known, the complex may 
be determined. Analytically, they give five equations, from which the five 
constants of the complex may be determined. Geometrically, four of them are 
cut by but two circles, conjugate with respect to the complex. A different four 
give another pair of conjugates. Two pairs of conjugates are sufficient to fix the 
pole of any sphere, or the polar of any point, for every circle cutting a pair of 
conjugates belongs to the complex, and of the circles cutting a pair of conjugates, 
one in general passes through any point, or lies on any sphere. The polar or 
pole is fixed by the two circles of the complex passing through the point or 
lying on the sphere. 

Three spheres and their poles being known, the complex is determined pro- 
vided not more than one of the circles of intersection belong to the complex 
i. e., not more than two of the poles lie upon the circles of intersection. For 
the conjugates of the circles of intersection are the circles determined by the 
poles of the corresponding pair of spheres. In the exceptional case two of the 
circles coincide with their conjugates, and the configuration determines but one 
pair of conjugates. 

The general case may be treated as follows : Given three spheres 8', S", 8'" 
with poles P', P", P'", we seek the polar of a point P. Draw the three spheres, 
(P, P', P"), (P, P', P'"), (P, P", P"). They cut the three circles of inter- 
section, (8' 8"), (8', 8'"), (8", #'"), in the points P x , P 2 , P 3 respectively. 
These points are the poles of the spheres respectively, for ( P' , P") and ( 8', 8") 
are conjugate circles, etc. P x , P 2 , P 3 being the poles of three spheres passing 
through a same point P, all lie on a same sphere (P x , P 2 , P 3 ), polar of P, 
and therefore containing P . The reciprocal problem, given 8 to find its pole, 
may be similarly treated. The point ( 8', 8", 8'" ) is on the sphere ( P', P", P") , 
since the poles of all the spheres passing through a point, lie on a sphere which 
contains that point as pole. 

The following table shows these relations : 



Sphere 


Pole 


Spheres Containing Pole 


8' 


P' 


(PP'P") (PP'P'") (P'P'P'") 


8" 


P" 


(PP'P") (PP"P'") (P'P'P'") 


8'" 


P" 


(PP'P") (PP'P'") {P'P'P'") 


(A^s) 


P 


(PP'P") (PP'P") (P P"P'") 


(PP'P") 


A 


S' 8" (P 1 P a P a ) 


(PPT") 


A 


8' S" (P,P 2 P S ) 


(PP'P'") 


P* 


8" 8'" (P,P 2 P 3 ) 


(P'P'P'") 


(8' 8" 8'") 


8' 8" 8" 



30 C. S. FORBES: THE GEOMETRY OF CIRCLES 

The configuration * therefore consists of two groups of four spheres, the poles 
of each of which are the intersections of three of the spheres of the other group. 
If such a group be called a tetrasphere, the configuration consists of two tetra- 
spheres, each of which is both inscribed in and circumscribed about the other. 
For the poles of the spheres of each are the vertices of the other. 



CHAPTER V 

Linear Congruences of Circles and Problems upon the Assemblage 
A. The Linear Congruence 

37. A linear congruence consists of the circles whose coordinates satisfy two 
linear equations of the first degree, i. e., the circles common to two complexes. 
Let a pair of complexes be 

f A = lA. kPik = 0, 
(70) 

Upon a sphere there lie in general one pencil of circles satisfying A and 
another pencil satisfying B . The circle belonging to both of these pencils is in 
general the only circle on the sphere belonging to the congruence. Reciprocally, 
through a given point there passes in general one circle of the congruence. 

38. The given complexes A and B may be replaced by any two of the pencil 
of complexes 

\ A + \B = (\,\ arbitrary) . 

Of the complexes of a pencil, two are special, for there are in general two solu- 
tions to the quadratic condition 

(71) a>(\A + \ 2 B) = 0. 

All the circles of a special complex intersect the directrix (see section 31). 
Hence the circles of the congruence, being common to the two special com- 
plexes, consist precisely of the oo 2 circles intersecting the two directrices of the 
special complexes of the pencil. The directrices are called directrices of the 
congruence. The circles passing through any point of one directrix, and inter- 
secting the other, lie on a sphere of which the point is the pole with respect to 
each of the complexes of the pencil, since the directrices are conjugates with 
respect to each of the complexes of the pencil. Hence the directrices of a con- 



Cf. Mobius, Uber Figuren im Baume, Crelle'sJournal, Vol. 10, p. 324. 



ORTHOGONAL TO A GIVEN SPHERE 31 

gruence may be defined * as the {"£**•} of the {$££} whose {$£,} are the 
same with respect to every complex of the pencil, of which the congruence is 
the common part. Any two circles conjugate with respect to a complex may be 
taken as directrices of a congruence belonging to the complex. Again, the 
directrices may be defined as, the locus of points {envelope of spheres) through 
which oo circles of the congruence pass. The equation of condition 

(71) <o(\A + \B) = Q 

may have two real roots, giving a congruence with real directrices, two imagi- 
nary roots, giving imaginary directrices, or two coincident roots, in which case 
the directrices coincide, the double directrix belongs to the congruence, and if a 
bilinear relation be set up between the generating points and the generating 
spheres of the directrix, then the assemblage of circles obtained by taking all 
and only the pencils that are contained on the spheres corresponding to the points, 
constitute the congruence.! In case (71) gives an indeterminate solution for 
\ : \ , the congruence has an infinity of directrices, which constitute a pencil of 
circles. The congruence consists of the circles of the point common to the cir- 
cles of the pencil, and of the circles of the sphere upon which the pencil lies. 

The theory of the congruence of circles is seen to be identical in form with 
those of the congruence of lines, and of the congruence of planes in point 
geometry of four space. J 

38. We now turn to certain special problems relating to the complex, and 
first find the condition that a complex shall contain a given congruence. Let 
the complex be given by 

and the congruence by 

£■»«!»« = 0, TC ikPik =0. 

The condition is, that the complex shall be one of the pencil of complexes 
defined by 

-Z(\B iJc + \C Vc ) Pik = Q, 

i. e., for some value of \ : X 2 the six equations 

\B ile + \C ik ^A ik 

shall be consistent. This is equivalent to the four independent conditions 
given by 



* Cf . Pluckee, Neue Geometrie, p. 78. 

fCf. Koenigs, La Geometrie Eeglee, chap. 3, Annales---Toulouse, 1892; Pluckee, 
Neue Geometrie, p. 62. 

t See references to Pluckee, Koenigs, Keyseb, etc. 



LIBRARY I) 



32 



C. S. FORBES : THE GEOMETRY OF CIRCLES 



(72) 



0. 



A complex may be subjected to five conditions. There are therefore oo com- 
plexes containing a given congruence i. e., the complexes of the pencil whose 
directrices define the congruence. 

The condition that a complex may contain a given circle p' is evidently 

(73) £4 fti 4 = 0. 

Similarly a complex will contain a given pencil 

\Pik+\P"i7c 

if it contains the generating circles of the pencil, i. e., 
The systems defined by the equations 

S5; A =o,...E5» A =o <»S4), 

belong to. the complex, if 

(74) 



giving (6 — n ) independent conditions. In the case n = 4 , the four equations 
above, and the quadratic identity define two circles, which accounts for the 
6 — 4 = 2 conditions. 

B. The Surface with Circles as Generators Formed by the Intersection of 
Two Congruences 

39. Two congruences of a complex of lines intersect in a ruled surface, the 
hyperboloid of one sheet, having two systems of straight-fine generators. In sec- 
tion 24 (p. 23) we set up a one-to-one correspondence between the lines of space 
and the assemblage of circles orthogonal to a given sphere. The analytical condi- 
tion that two circles intersect is precisely the condition that the two corresponding 
lines intersect. Hence by analogy we derive the properties of the surface formed 



ORTHOGONAL TO A GIVEN SPHERE 33 

by the intersection of two congruences of circles of a complex. Consider two 
congruences of a complex, as defined by two pairs of circles p x , p[ and p 2 , p' 2 , con- 
jugate with respect to the complex. Any circle intersecting two conjugate circles, 
belongs to the complex, and to the congruence of which the two conjugate cir- 
cles are directrices. Of these oo 2 circles, oo 1 circles intersect any other circle, 
and two of these oo 2 circles in general intersect any pair of circles. If this lat- 
ter pair are conjugate, with respect to the complex, all the circles of the complex 
which intersect one intersect the other also. The theorem follows : 

oo 1 circles of a complex intersect two pairs of circles conjugate with respect to 
the complex, i. e., two congruences of a complex intersect in a configuration 
composed of oo 1 circles of the complex. 

These circles generate a surface such that through every point on the surface 
pass two generating circles. The circles are arranged in two systems constitut- 
ing a net of circles. No two circles of the same system intersect, but every 
circle of the one system intersects every circle of the other system. A sphere 
which contains one circle of the surface contains a second circle of the surface, 
and may be considered as tangent to the surface at the point-pair in which the 
two generators intersect. Since no two circles of the same system intersect no 
sphere can contain two generators of the same system. 

In general two congruences do not belong to a same complex. The condition 
is that one of the generating complexes of one congruence, shall be a generating 
complex of the other congruence. Three complexes, however, intersect in a sur- 
face of the above character. Its equations are therefore given as three equa- 
tions of the form "E,A ik p jk = . 

C. Radius of a Circle, Circles in Involution, Degeneration of S 5 
We treat in this division several special problems upon circles and the assem- 



40. The radius p of a sphere 

£a.as.= 

is 

(75) M) 

Every sphere f passing through a circle p ih is given by the equation 
(76) £(>*>* + A*p / wK='0, 

where a , /3 are two of the indices 1 , 2 , 3 , 4 , and A and /x are arbitrary. The 
radius of this sphere is 

* Daeboux, Theorie des Surfaces, yol. I, p. 227. 

fKoENlGS, Contributions, etc., Annales--- Toulouse, 1888, p. F 6. 



34 C. S. FORBES: THE GEOMETRY OF CIRCLES 

XOPai + ^lV) 2 



(77) 



[xzt + ^tJ 



When this is a minimum the radius of the sphere will be equal to that of the 
circle. Its value will be 

IX- 

(78) P 2 = 



?(?t) ! 



When p = 

(79) E*!-H(iO-.o. 

This is a complex of the second order, and consists of the points of S. , regarded 
as circles orthogonal to S R . Each of the points is taken infinity times and in 
fact is a degenerate pencil of circles, having as axis the radius of S 5 through 
the point. 

When p = oo, 

(so) z(z$;) 2 =o. 

This is therefore the equation of the straight lines passing through the center of 
S 5 , each taken oo times, i. e., regarded as a degenerate pencil of circles orthog- 
onal to S s and of infinite radius. If 



(i,j=l, •••4). 



P'iU = HPikPiU 

has been shown by Koenigs * to play an important role in the theory of circles 
in space. Let there be given two circles p ilt andp^. If thorough one of them 
a sphere can be drawn orthogonal to the other, then through the second a 
sphere can be drawn orthogonal to the first, and the circles are said to be in 
involution. The analytical condition is found to be 

(82) 3(p, P ') = 0. 







B\ 


= R\ = 




equation (80) degenerates into 


2>, 


= 0. 


41. 


The polar 


of the form H 


(ph 




(81) 




H(p, P ') = 


24- d 


>(P) 



Koenigs, Contributions, etc., Annales--- Toulouse, 1888, p. F 9. 



ORTHOGONAL TO A GIVEN SPHERE 35 

The transition from the theory of circles in space, to the theory of circles 
orthogonal to a given sphere, consists solely in a proper choice of fundamental 
spheres, and the limiting of the range of i , k to the values 1 , 2 , 3 , 4 , instead 
of 1,2, 3, 4, 5. If one of the circles, p ik be fixed, the circles in involution 
with it form a special complex, since they are given by the equation 

(83) 2»,,= °> 

[see (82)] , and the coordinates of the complex are the coordinates of a circle 
[see (60)]. The quantities p' ilc are not the coordinates of the directrix of the 
complex, regarded as an envelope of spheres ; but writing in full equation (83), 

PuPu + PnPn + P'uPu + PnPn + PuPa + PsiPu = ° > 
and the condition that the circles p' ik and p ik intersect [see (22)] , 

pLPu + P'uPiz + PnPu + P'uP* + PnPi2 + PuPu = ° » 
it is readily seen that the coordinates ^^ are proportional to the coordinates q' ik , 
which give the directrix as the locus of points [see (19)] . Hence the theorem : 
The circles in involution with a circle p' ik form a special complex whose direc- 
trix is given by coordinates q' ik proportional to the coordinates p' ik . 

The same reasoning applies to the coordinates A ik of any special complex 
[see (60)] 

T.A ikPik =0. 
The theorem follows: 

The coordinates A. k of a special complex are proportional to the coordinates 
q ik of the directrix. 

42. The angle between two spheres 

la.a;. = 0, and S^ a3 J = 

is given by 

(84) y = cos -i^ = £ a ' 5 ! * 

If the spheres are tangent externally or internally 

Squaring and expanding we obtain 

(85) Z(«A-«A.) 2 = Zi4 = o. 

Conversely, if 

the spheres are tangent. Two spheres, however, can only be tangent at a point 
on the sphere S s . Hence (85) is the equation of the points of S s , regarded as 

* Daeboux, Theorie des Surfaces, vol. I, p. 228. 



C. S. FORBES: THE GEOMETRY OF CIRCLES 



circles orthogonal to S.. Each point is taken oo times, since a pair of spheres 
may be tangent at any point on S s in oo positions. If the quantities p ik are 
real, they must all equal zero, and their ratios be indeterminate. These results 
are identical with that of (79). 

43. The system as a whole may degenerate in two ways, — the radius of S s may 
become zero or infinity. In the former case S s becomes a point. Each of the 
oo 3 spheres and oo* circles passing through the point may be regarded as orthog- 
onal to it. The inside elements of all the point-pairs coincide and this point 
taken oo 3 times is regarded as related by inversion to every point of space. The 
formulae of inversion of course become indeterminate. 

Secondly, if the radius of S 5 becomes infinite, S 5 becomes a plane. The oo 3 
spheres and oo 4 circles orthogonal to the plane, all have their centers in the 
plane. Points on opposite sides of the plane, and symmetrically placed, form a 
point-pair, their axis being perpendicular to the plane, just as the axis of a point- 
pair in the general case is a radius of the sphere and therefore perpendicular to 
the surface of the sphere. 

CHAPTER VI 



44. The 



Families or Spheres 
leres for which the set of ratios 



is constant are said to form a family of spheres. We shall show that families 
of spheres exhibit many of the characteristics of families of parallel planes in line 
geometry. 

The pole of the sphere 

'£a.x i = 

x 2 =- A a a 1 + A 23 a 3 - A a a^ 
x s =- A l3 a x - A 23 a 2 + A u a t , 
x,= - A x ,a x + A i2 a 2 - A 3i a 3 = A . 

a 1 cc 1 + a 2 x 2 + a 3 x 3 = 

belongs to the same family and is orthogonal to the fundamental sphere S t . 
The pole of (87) is 

x[= A 12 a 2 +A 13 a 3 , 



is, [see (43)], 


(86) 


The sphere 

(87) 



(88) 



x' 2 = — A l2 a^ + A 23 a 3 , 

x' 4 = — A u a y + A i2 a 2 — A u a 3 = A. 



ORTHOGONAL TO A GIVEN SPHERE 

From (86) and (88) we obtain 

f x 1 — x' l = A u a 4 , 
(89) J x 2 -x' 2 = -A i2 a 4 , 



37 



Dividing the several terms of these equations by 

x 4 = x 4 = A, 
we obtain 











x i 


x[ 


A u a 4 










X A 


x i 


A ' 


(90) 








x 4 


< 

x 'i 


A i2 a 4 
A ' 

A 3i a i 

A ' 


whence 














(91) 


X 


< 


— x 4 a 


1 


x 2 x 4 — 
— A 


x 4 x 2 x 3 x 4 x 4 x 3 






4 


~ A- 



"14 J ~ l 42 ""34 

If we regard x t as variable, it is seen that these are the equations of a circle, the 
locus of points x { the poles of the spheres of the family 

a t : a 2 : a s . 

For since the equations are independent of a 4 , they are satisfied by the coordi- 
nates x i of the pole of any sphere of the form 

(92) a x x x + a 2 x 2 -f a 3 x 3 + \x 4 = 
Similarly the poles of the spheres of the family 

(93) \ x r + b 2 x 2 + b 3 x 3 + Xx 4 = 
lie on a circle whose equations are 



( A arbitrary ) . 



(94) 



x iVi - <Mi _ x 2Vi- x iV2 



where the point y\ is the pole of the sphere 
b 1 x 1 + b 2 x 2 + b 3 ', 



0. 



We shall now show that the two circles (91) and (94) lie on a same sphere. 
Expanding we have as the equations of the spheres, S and S', defining (94), 



(95) 



A *y'& - A uy>2 + ( A *y[ + AM x 4 = °. 

A *V't*x - A uV>3 + ( A uVs ~ A zM x ± = °- 



38 C. S. FORBES: THE GEOMETRY OF CIRCLES 

Similarly from (91) we obtain, 

A i2 X 'i X l - A U X 'i X 2 + ( A 42 X 'l + AXK = - 



(96) 

' A U X 'i X l ~ A U X >3 + ( A U X 3 ~ AXK = - 

In order that the circles may lie on a same sphere, it is necessary and sufficient 
that one of the pencil of spheres 

S-\-\S' ( 2. arbitrary ) , 

defined by (95), should be identical with one of the pencil 

S" + /J>S'" ( fi arbitrary ) , 

defined by (96), for some values of X and /*. That is, equating ratios of coeffi- 
cients 

A u.y\ + xA 3 ,y'i _ - A i 2 x ' i + ^^ 
y't A u < A u 

i^y'i + A M + H A uy' s - AM 
y'Au 

(A a x[ + A U X' 2 ) + f*( A u X 3 ~ A U X 'l) 



(97) 



The second of these equations of condition requires that 

X = /A. 

Substituting this value in the first equation, it is seen to be satisfied identically 
whatever the value of \a . There remains but one equation and \a may be taken 
so as to satisfy it. The theorem follows : 

The loci of the poles of two families of spheres are two cwcles lying upon 
a same sphere. 

We call such circles diametral circles. 

45. From the last of equations (97) we obtain 

_ y'A x 'i A 42 + x 2 A u) - x 'Ay'i A 42 + y'2 A u) 

x' i (y' 3 A u -y' 1 A 3i )-y' i (x' 3 A u -x' 1 A 3i )' 
which reduces to 

A u <o(A)p l3 - A i2 <o(A)p 23 

A U<°( A )Pl2- A U<*( A )P2Z 

_ A liPl3 ~ A i2i°23 . 
A uPl2- A 3iP2 3 ' 

When the complex is special, /* is indeterminate. Otherwise \a is a function of 
A u , A^ and A Si and of a x , a 2 , a s and \, b 2 , 6 3 , and changes as these latter 
quantities change, i. e., is different for different families. It follows that the 



ORTHOGONAL TO A GIVEN SPHERE 6V 

circles associated with different families do not all lie on a same sphere, hut 
have a relation similar to that of parallel straight lines in space. Any two 
of these lines lie in a plane just as any two of the circles lie in a sphere, but 
in general three lines or three circles do not lie in a same plane or on a same 
sphere. Plucker * has shown that the locus of the poles of a system of paral- 
lel planes is a straight line ("Durchmesser"), and all these diameters are paral- 
lel. A family of spheres in the circle theory is seen to correspond to a system 
of parallel planes in the line theory. 
46. There are oo 2 families, since 



involves two parameters. There are oo 2 diametral circles, since in general 
each family defines its diametral circle uniquely. 

Upon each of the generating spheres of a diametral circle, lie oo diametral 
circles. 

Tliere are oo 2 circles, not lying on the common sphere of two given diame- 
tral circles, out co-spherical with each of them separately. These circles are 
co-spherical with every diametral circle. 

For consider a circle given by a sphere of the pencil (95) 

(98) S+XS', 
and by a sphere of the pencil denned by (96) 

(99) S" + fiS'". 

Let the equations of any other diametral circle be 

- A ^ X 1 - A U Z 'i X 2 + ( A iA + A lA) X i = > 



(100) 

A zA x i - AA x 3 + ( A iA - A u z 'i) x i = °- 

The condition that the circle defined by spheres (98) and (99) shall inter- 
sect the diametral circle (100) is the vanishing of the determinant of the coeffi- 
cient of these four spheres i. e., the condition that they have a point in common, 

— pA Q x'i — pA^x'^ — A^x\ pA i2 x[ etc. 

-p^yl -P A uy'± ~ A uv', P A ^y'x Qt(i - 

A„z': — A„z' A,X etc. 



(101) 



Taking the factor — A u from the second and third columns and replacing it by 



*PLiJCKEE, Neue Geometrie, etc., pp. 35 et al. Also M5BIUS, Crelle's Journal, vol. 10, 
pp. 329 et al. 



40 



C. S. FORBES : THE GEOMETRY OF CIRCLES 



A..„A-„ 



J. 34 a^etc. — pA a x[ A 3i x[ pA i2 x' x etc. 

A 3i y ■[ etc. -pAuX't A^ 
— A a %\ -A i2 z[ 





Adding third column to second, we obtain 

A Zi X 'i - P A i2 X 'i A zA - P A *2 X 'i A U X 'l 
A *y'l- PA-aV'l A M X 'i - P A 42 Z 'i A uV'i 



A..~A.~ 



A, 



= 0. 



"34 ~i "34 ~4 

The theorem follows : 

A circle co-spherical with two diametral circles, but not lying on their 
common sphere, is co-spherical with every diametral circle. 

In line geometry every line co-planar with two diameters but not in their 
common plane, i. e., parallel to them, is a diameter. Hence by analogy each of 
these oo 2 circles is a diametral circle. 



CHAPTER VII 

Transformations of the Assemblage 

A. Generalization of Coordinates j ik 

47. "The coordinates 7 i7£ " of section 11, "admit of generalization.* We 
know from the theory of forms, that the new variables v { in the transformation, 

P%* = C a, i v i + G iu, 2 v 2 + ■ ■ ■ C ». 6 v e » 
where the modulus is not zero, are connected by a homogeneous quadratic identity 

O(*)ss0, 

where O(y) is the transformed of (o(y). Moreover the polar form ©(7, 7) 
of (o(y) is converted by the same transformation into the polar form, 

of £l(v). It is well known that o)(y), regarded as a quadratic form, has a 
non-vanishing discriminant, and that it is possible to find a linear transforma- 
tion which will convert this form into any quadratic form X2(y), whose discrim- 

*See KEYSEB, Plane Geometry, etc., p. 310. KOENIGS, La Giomrfrie Eiglin. 



ORTHOGONAL TO A GIVEN SPHERE 



41 



inant does not vanish. Accordingly we may employ for homogeneous" circle 
"coordinates any six variables v. connected by the quadratic identity Q, (v), 
where £l(v) has a non- vanishing discriminant. Hence the condition that the" 
circles "p. and v'. shall intersect in a" point pair "or lie on a same" sphere 
"is that 



B. Transformation of Fundamental Spheres 

48. We now consider the general transformation of the fundamental spheres 
S { . The equation of a sphere in space in pentaspherical coordinates is 

(102) S = a x x x + a 2 x 2 + a s x s + a i x i + a 5 x 5 = , 

where 

(103) 



px. 



P. being the power of the point x t with respect to the sphere S t , and R. being 
the radius of 8. . Consider the linear transformation 



(104) 



a x = a n a x + a u a 2 + a n a % + a^a^ + a 15 a. 

a 2 = a 2i a i + a 22 a 2 + a 23 a 3 + « %i < + « 2 5 «. 
«3 = Si ! + a 32«2 + a 33«3 + «34 «4 + So «. 

a 4 = a^ + a 42 a 2 + a iZ a' z + a,^ + a^a, 

°i = a oi a i + a 52 a 2 + a 53 a 3 + ««< + ^55 °i 

which involves twenty-four parameters. The sphere is transformed into 



(105) 
where 



& = £ ®i S a ,-X = Z «* E «»»< = £ «X = o 



EVi- 



We now seek the conditions that a point P (xyz) represented by x { , shall be 
represented by x' k . That is 

(106) K = §' (*=l,--,B), 

where P' k is the power of the point P ( xyz ) with respect to a new fundamen- 
tal sphere S' k , and R' k is the radius of that sphere. Expanding x i we obtain 

(107) x^ B - , 



42 C. S. FORBES : THE GEOMETRY OP CIRCLES 

whence 

EvrSlC + ^ + O 



X 

(108) 



Write oc^fH. = /3. 4 , and divide numerator and denominator by ]£,-/3. 4 , 

(109) *; = j— i^* — . 

Adding and subtracting 

( j (Z/3,) 2 

in the numerator, it becomes of the form 

\ ) k n •> 

where if x' k is of the form (106) 

pn 2 =r- (p arbitrary ) , 



(112) 



(IU (ZPJ 2 



This imposes one condition upon the quantities /3 j7£ , i. e., on a ik . Hence in 
general that the five quantities x' le may represent the same point P(xyz) that 
x i represents, it is necessary and sufficient that the quantities a ik be subjected to 
four conditions of the form (112), and the transformation is equivalent to a 
change of fundamental spheres. 

We next impose the condition that the sphere S 5 may remain unchanged, i. e., 
that the system of orthogonal circles which compose the space may remain 
unchanged. It is necessary and sufficient in order that 

x 5 — X i ' 
that 
(113) a 15 = a 25 = a z . = a i5 = and a 55 = 1 . 

Impose further the ten conditions that the five spheres, S' { , shall be mutually 
orthogonal. Since the four fundamental spheres 

S 1 , S 2 , o 3 , >S 4 



a 1 = a n a 1 


+ • 


■■ a u a , 


a 2 =a 2l a[ 


+ • 


■■ a u< 


a 3 =a sl a[ 


+ • 


•«34< 


. a i = a il a' 1 


+ • 


■■ a u< 



ORTHOGONAL TO A GIVEN SPHERE 43 

are orthogonal to S' 5 = S 5 , it follows that every sphere of the form 

(114) 2>X = o 

is orthogonal to S s . The theorem follows : 

The system of circles orthogonal to a given sphere may he transformed into 
themselves referred to form new fundamental spheres of the system, by a gen- 
eral linear transformation of the quantities a { , the twenty-four parameters of 
the transformation being subject to eighteen conditions. 

49. As a special case let the sphere S 5 be unchanged, and the quantities 
a { ( i = 1 , 2 , 3 , 4 ) be transformed by the relations 



(115) («u- 



This involves fifteen parameters. S becomes 

4 4 4 4 

(116) S' = £ x. X) a. k a' k 3 5X £ ct. k x. = £ < x' k = . 

\i Ik 1* U k 

If x' k and x. represent the same point P(xyz), the quantities, a ik , are sub- 
jected to three conditions. If the new fundamental spheres £; , ( i = 1 , •••4), 
are orthogonal, six more conditions are imposed. As before there are left six 
degrees of freedom. 

C. Invariance of a> (A) 

50. We now show the in variance of the fundamental form, co(A'), of the 
complex. Consider the complex 

(H7) *A ikPik = Q. 

A linear substitution 

(118) a i = a. l a' 1 +a. 2 a 2 +a. 3 a , 3 +a. i a' l (« = l,---4) 

has been shown to be equivalent to a change of fundamental spheres. Let 

(H9) «;=I>X> &,-Z>a»;- 

k k 

By this substitution 



l)6COm6S 


Pik = a A- a k h 


(120) 
where 


Pik = Z) hl-jrP'jr 


(121) 


p' Jr =a'.b' r -a' r b'., l. kJr =c 



44 C. S. FORBES : THE GEOMETRY OP CIRCLES 

Substituting these values in the equation of the complex, we obtain 

(122) -LA'. r p' jr = 0, 

where 

(123) i;-s^- 

ik 

From these form the invariant 

+ ">,>( W' W A 3 4 42 + »*(V' W A^2 



(124) 



<*21 <*22 a 23 a 2 

a_ a.. ol, a, 



©(4) = Aa)(^L), 



For terms of the first two types vanish identically, and the coefficients of the 
last three terms are all equal to A . Since co ( A ) is in general not zero, only 
four of the coefficients of a non-special complex can become zero through any 
transformation, and the two remaining must not have a common subscript. The 
complex may take any of the three forms 



(125) 



A 12Pl2 + A LP3i= -- 



A uPu + A 23P23= - 



There are still two degrees of freedom among the coordinates of the transfor- 
mation. It will be remembered, however, that the reduction of the complex of 
lines to its simplest equation, is of such a character that the complex may 
undergo symmetrical translations and rotations with respect to its axis, the equa- 
tion remaining in its simplest form. Hence in both the cases of line and circle 
complexes there remain two degrees of freedom. 

51. Since the elements of a point-pair are simply interchanged by the formulae 
of inversion, it follows that every point-pair, sphere or circle of the assemblage 
is invariant under this transformation. The theorem follows : 

Every configuration of the assemblage is invariant under the transforma- 
tion by inversion with respect to the fundamental sphere S 5 . 

D. Projective Transformation by Means of Complexes 

52. We have * seen that any linear complex C sets up a one-to-one corres- 
pondence between the spheres and the point-pairs of the assemblage. Let p and 

* Compare Keyser, Plane Geometry, etc. 



ORTHOGONAL TO A GIVEN SPHERE 45 

p' be two circles conjugate with respect to C. Each is the {iocu e J ope } °f the 
{po^} of the {feres' 1 " 3 } of the other. Consequently if S x , S 2 , S s , S A be four 
spheres of p, their poles P 1 , P 2 , P 3 , P i lie upon^>' and 

(S 1 S 2 S 3 S i ) = (P 1 P 2 P 3 P t ), 

i. e., every complex sets up a projective transformation between the spheres and 
point-pairs of each pair of circles conjugate with respect to it, and between the 
spheres and point-pairs of each self -conjugate circle, i. e., each circle belonging 
to the complex. 

If p be a circle common to two complexes, and S l , • • • , # 4 be any four of its 
spheres whose poles with respect to one of the complexes are P l - • • P 4 , and 
with respect to the other P[ , • • • , P' 4 , then 

(P 1 P 2 P 3 P i )=[(S 1 S 2 S 3 S i )] = (P' 1 P' 2 P' 3 P' i ). 
Similarly 

(44*.^)- [(PxP.p.p.M-C^^,^). 

The theorem follows : 

Any two linear complexes set up a projective transformation between the 
{fJint e - s P air S } of each of the circles of their common congruence. 

Accordingly any circle of the congruence may be regarded as two superim- 
posed {Sanies 8 } °f { po^nt-^airs } j associated one with each of the complexes which 
define the congruence. Any circle of the congruence is common to all the com- 
plexes of the pencil of complexes 

T,(\O 1 + \C 2 )p = 

which define the congruence. Let p be any circle of the congruence. Its point- 
pairs and spheres are transformed by every complex \ : \ = m of the pencil. 
Since the director circles of the congruence are conjugate with respect to every 
complex of the congruence, it follows that the {$£■} of the {P°ieres airs } common 
to p and to one of director circles, are the same with respect to every complex 
of the pencil. Hence these {spheies airs } w ^l be tne ^ oc ^ °^ every projective trans- 
formation of the {^pheres airs } °f P by means of any pair, m 1 and m 2 , of the com- 
plexes. 

Let rap m 2 , m s , m i be any four complexes of the pencil, and P 1 , P 2 , P 3 , P 4 
be the corresponding poles of any sphere S of p , then 

(m 1 m 2 m 3 m 4 ) = (P 1 P 2 P 3 P 4 ). 

In particular if S t and S 2 be the special complexes of the pencil, 

(m 1 S 1 m 2 S 2 ) = (P 1 P dl P 2 PJ = k, 

since P 1 and P 2 are any pair of corresponding elements and P di and P do are 
the foci. Hence the 







P,- 


P, 


P,~ 


P, 






P,~ 


■**' 


' P«>- 


P*i 


oo . 


, and P t 


h = 0, 


We have 










P* 


-P, 


:h. 



46 C. S. FORBES : THE GEOMETRY OF CIRCLES 

mon to a pair of complexes by the complexes, are formally identical. Recipro- 
cally a transformation is effected among the complexes of a congruence by the 
{pole! 3 } °£ an y {spher"e pair } of a circle of tne congruence. The foci are the special 
complexes. 

53. The equation of the transformation of the point-pairs of a circle with 
respect to a pair of complexes assumes its simplest form when referred to the 
foci. We have the relation 

{P 1 P dl P,P d2 ) = lc; i. e.,* 
Let P A 



P 1 = (1-A)P 2 , 

which is the required transformation. Similarly 

(S 1 S <k S 2 S dli ) = h, 
and 

^=(1 -&)&,. 

Hence the point-pair and sphere transformations effected upou the circles 
common to two complexes by the complexes, are formally identical. 



CHAPTER VIII 
The Complex of Second Degree 

A. Tangent and Polar Linear Complexes 

54. The circles whose coordinates p ik satisfy a homogeneous equation of the 
second degree form a complex of the second degree. The equation may be 
written 

^2 = A VmPl + A lZ\zP\Z + A UliPli + A 2mPzi + A 4242Pi2 + A U3iPzi 
+ 2^1213^12^3 + 2A 13uPlzPu + 2 ^1223^12P23 + 2A t*aPuPa 

(126) + 2A l2SiPl2 p Bi + 2A 13liPl3 Pli + 2A imPu p 23 + 2A im p uPi2 

+ 2^1334^13^34 + 2A imPuP2 3 + * A imPuPu + 2A imPuPu 

+ 2^2342^23^42 + 2^2334^23^34 + ^ A 4mPaPu = ' 

* Notation of Doehlemann, Projehtive Geometric 

t The P's are of course to be replaced by their coordinates if a metrical interpretation is 
desired. 



ORTHOGONAL TO A GIVEN SPHERE 47 

Let the polar form of C 2 be 

(127) tf 2 (^P)-*Z|^-i£||p,. 

Two circles, p ik and p' ik , which satisfy the equation 

(127) C 2 (p,p') = 0, 

are said to be associate with respect to C 2 . Holding p' ilc fixed and letting p ih 
vary, (127) is the equation of a complex. The theorem follows : 

All the circles associate with a given circle with respect to a complex of 
second degree, form a linear complex. 

The circle is called the pole of the complex, and the complex is the polar of 
the circle. It follows at once from (127) : 

If p' ik is in the polar of p ilc , then p ilc is in the polar of p' ilc . 

If p'. h satisfies C 2 it is among the circles of its polar complex, for the polar 
equation reduces identically to C 2 (p') when 

The polar complex is then said to be tangent to the complex of second degree. 
A linear complex does not in general possess a pole with respect to a given 
complex of second degree. For while the six equations 

(128) > A '*= S -W> 

define in general the ratios of the six quantities p' ilc uniquely, these quantities p' iJe 
do not in general satisfy the identity 

a>(p') = 0, 

and hence are not the coordinates of a circle. There are in fact oo 5 complexes, 
and but oo 4 circles. Two, however, of the complexes of a congruence possess a 
pole. For let the congruence be defined by the pencil 

(129) S(M« + M»)Pa-o- 

The seven equations 

(130) Mi+V^-^. 

co(p)=0 

involve seven unknown quantities, p ik and \ : \ , and are in general capable of 
two solutions. 

If a circle is associate with respect to two circles p' iJc and^"^, it will be associ- 
ciate with every circle of the pencil defined by these two circles. For 

(131) C 2 (p, \p' + \p") = \C 2 (p,p') + \C 2 (p,p") = 0. 



48 C. S. FORBES : THE GEOMETRY OE CIRCLES 

B. Intersection of a Pencil of Circles with C 

55. A pencil has two circles in common with a complex of second degree. 
For the equations of a pencil are 

(132) Pu-\P'l2 _ Pl3-\Pl 3 _ __ Pzi-\P'u _ x 

P"l2 Pl3 Pi "" 

The circles common to the pencil and C 2 are found by eliminating p. Jc between 
these equations. We find 

(133) c 2 ( P ")xi + x 2 x^ P ': 7 8 -^P + x\c 2 { P ') = o. 

This is a quadratic equation in \:\, and is satisfied by two values of this 
ratio, i. e., the pencil and complex have two circles in common. If these two 
circles coincide, the pencil is said to be tangent to the complex. The condition 
that (133) has equal roots is 

(134) (^P^^^) -±C 2 (p")C 2 { V ') = Q. 
If C 2 {p' ) = 0, the condition becomes 

(135) EA^-1 
The theorem follows : 

If p' ik be a circle of a complex of second degree, the circles p" ik which 
generate with p' ik pencils tangent to the complex of second degree form a 
linear complex. 

A similar theorem evidently holds for p" ik . This complex is seen from its 
equation to be the tangent complex. Reciprocally : If p" ik be a circle not belong- 
ing to a complex of second degree, the circles p' ilc of that complex, which with 
p" ik generate pencils tangent to the complex, belong to a linear complex polar 
to p" ik with respect to the complex of second degree. 

56. Attention is called to the similarity between the theory here developed, 
and the theory of polars with respect to a conicoid. The following elements 
enjoy corresponding properties in this regard : 



Circle Geometry. 


Point Geometey. 


Circle 

Pencil 

Linear complex 

Complex of second degree 


Point 
Line 
Plane 
Conicoid 



ORTHOGONAL TO A GIVEN SPHERE 49 

C. Generalization of Pliicker 

57. The deductions of section 54 are not strictly general. Following 
Plucker * we may proceed as follows. Every circle whose coordinates satisfy 
a complex of second degree C 2 also satisfies the equation 

C 2 + Xco(p) = ( X arbitrary ) . 

The polar form of this is 

(136) C 2 ( P ,p') + \co(p,p'). 

Hence to every circle there are an go of polar complexes, and to every complex 
there is a pole. Likewise if p' i!c is one of the circles of C 2 , there are co com- 
plexes tangent at p\ k . Since co (p , p ) = is special, p' ik is one directrix of the 
congruence defined by (136). The other directrix is the conjugate of p' iJc , and 
is the same with respect to all the complexes of the congruence. The second 
special complex is given by 

d_C.dC, dC.dC, dC^dC^ 

^ ) 2 G i(P') 

If C z (p' ) = , both directrices coincide, and 

A, = oo or -^ . 

Circles of C 2 for which A, = $ are called singular circles of the complex. A 
treatment of this subject would exceed the limits of this paper. The theory of 
circles orthogonal to a given sphere might be pushed much further, but in 
showing the similarity of the theory to that of line geometry, and the develop- 
ment of its fundamental problems, we have accomplished our object. 



BIBLIOGRAPHY 

The following articles treat of topics more or less intimately associated with 
the subject of the present paper. Those which have been of particular value to 
us are marked with a ( * ) . 

Catlet. On the Six Coordinates of a Line. Collected Mathematical 

Papers, vol. VII, p. 66. 
Sixth Memoir on Quantics. Op. cit., vol. II, p. 561. 
Cosserat. Sur* le cercle considere comme element generateur de Vespace. 

Annales de la Faculte des Sciences de Toulouse, vol. Ill, 

1889, p. E 1. 

* Pluckee, Neue Geometrie, pp. 287-296. 



50 C. S. FORBES: THE GEOMETRY OF CIRCLES 

Darboux. Sur une classe remarkable de courbes et de surfaces algebriques. 
Theorie * des Surfaces, vol. I, book II. chap. 6. 

Sur les relations entre les groups de points, de cercles, et de spheres dans la 

plan et dans V espace. Journal de Liouville, 2 nd series, vol. 1, 1872. 

Demartres. Sur les surfaces a generatrice circulaire. Annales del' 

Ecole Normal e, 3 rd series, vol. II. p. 123. 
Enneper. Die cyhlischen Fl'dchen. Zeitschrift fur Mathematik und 

Physik, p. 393, 1869. 
Ketser. The* Plane Geometry of the Point in Space of Pour Dimensions. 

American Journal of Mathematics, vol. 25, No. 4. 
Klein. Tiber der sogenannte Nicht-Euclidische Geometric. Mathemat- 
ische AnnaleD, vol. 4, p. 573, vol. 7, p. 531, vol. 37, p. 544. 
Einleitung * in die Pcohere Geometrie, vol. I. 

Uber Liniengeometrie und metrische Geometrie. Mathematische 

Annalen, vol. 5, p. 257. 

Koenigs. Contributions* a la theorie du cercle dans V espace. Annales 

de la Faculte des Sciences de Toulouse, vol. II, 1888, p. F. 1. 

La* Geometrie Reglee. Op. cit., vol. 111,1889: vol. VI, 1892; vol. 

VII, 1893. 
Sur les proprietes infinitesimales de V espace regie. Annales de l'Ecole 
Normal, 1882. 
Lie. Tiber* Complexe, inbesondere Linien- und Kugel-complexe. Mathe- 
matische Annalen, vol. V, 1872, p. 164. 
Loria. Remarques * sur la geometrie analytique des cercles du plan. Quar- 
terly Journal, vol. XXII, p. 44. 
Mobius. Tiber eine besondere Art dualer Verh'dltnisse zwischen Figuren 
im Paume. Crelle's Journal fair Mathematik, vol. 10, p. 317. 
Pasch. Zur Theorie der linear Complexen. Crelle's Journal fair 

Mathematik, vol. 75, p. 11. 
Plucker. Neue * Geometrie des Raumes. 

On* a New Geometry of Space. "Wissenschaf tliche Abhandlun- 
gen, p. 469. 
Stephanos. Sur la geometrie des spheres. Comptes Rendus de l'Acad- 
emie des Sciences, vol. XCII, 1881, p. 1195. 
Sur une configuration remarquable de cercles dans V espace. Op. cit., vol. 

XCIII, p. 579. 
Sur une configuration de quinze cercles et sur les congruences lineares de 
cercles dans I 'espace, vol. XCIII, p. 633. 
Smith. * Solid Geometry. 
Snyder. Die Lieschen Kugel-geometrie. 
Study. Geometrie der Dynamen. 
Sturm. Line Geometry. 



*r* 



i^*i 






tm. 






tk< #• 



V,£W 



\N>- 01 




I 



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